Calculating intermodulation products and intercept points for circuit distortion analysis

ABSTRACT

A pertubative approach based on the Born approximation resolves weakly nonlinear circuit models without requiring explicit high-order device derivatives. Convergence properties and the relation to Volterra series are discussed. According to the disclosed methods, second and third order intermodulation products (IM2, IM3) and intercept points (IP2, IP3) can be calculated by second and third order Born approximations under weakly nonlinear conditions. A diagrammatic representation of nonlinear interactions is presented. Using this diagrammatic technique, both Volterra series and Born approximations can be constructed in a systematic way. The method is generalized to calculate other high-order nonlinear effects such as IMn (nth order intermodulation product) and IPn (nth order intermodulation intercept point). In general, the equations are developed in harmonic form and can be implemented in both time and frequency domains for analog and RF circuits.

BACKGROUND OF THE INVENTION

The present invention relates to the analysis of circuits generally andmore particularly to the distortion analysis of analog and RF (RadioFrequency) circuits by calculating intermodulation products andintercept points.

Second and third order intermodulation intercept points (IP2 and IP3)are critical design specifications for circuit nonlinearity anddistortion. A rapid yet accurate method to compute IP2/IP3 is of greatimportance in complex RF and analog designs. Conventionalcomputer-aided-design solutions for measuring IP2/IP3 are typicallybased on multi-tone simulations. For a circuit with a DC (DirectCurrent) operating point such as a LNA (Low Noise Amplifier), a two-tonesimulation is performed at two RF input frequencies ω₁ and ω₂ (usuallyclosely spaced). When the RF power level is low enough, signals atfrequencies ω₁−ω₂ and 2ω₁−ω₂ are dominated by second and third ordernonlinear effects respectively, and higher order contributions arenegligible compared to the leading-order terms. Thus, solutions at ω₁−ω₂and 2ω₁−ω₂ can be used as second and third order intermodulationproducts (IM2 and IM3) to extrapolate intercept points for IP2 and IP3.For a circuit with a periodic time-varying operating point such as amixer or a switch capacitor filter, a three-tone simulation at ω₁, ω₂and the LO (local oscillator) or clock frequency ω_(c) can be conductedand IM3 can be measured at frequency 2ω₁−ω₂−ω_(c) [8, 13, 3, 4].

Because of the low RF power setting, very high accuracy is typicallyrequired in order to obtain reliable intermodulation results.Particularly in three-tone cases, the numerical dynamic range has toaccommodate the large LO signal, the small RF signals, and the nonlineardistortions. Furthermore, a multi-tone simulation is generallyinefficient for IP2/IP3 measurements because, in addition to IM2 and IM3harmonics, this approach also resolves other irrelevant frequencies toall nonlinear orders. This additional overhead can be very expensive inlarge designs with thousands of transistors.

Essentially, IP2/IP3 calculation is a weakly nonlinear problem. It issubstantially concerned with only the leading second or third ordereffects. The fully converged multi-tone solution that contains everyorder of nonlinearity is generally unnecessary. A more efficient way isto treat both RF inputs as perturbation to the operating point solutionand apply 2nd or 3rd order perturbation theory to calculate IM2 or IM3at the relevant frequency directly. In this way, the dynamic range isreduced to cover just the RF excitations. The most commonly usedperturbative method for distortion analysis is the Volterra series [12,2, 7, 1, 11, 5, 16, 17]. However, this approach requires second andhigher order derivatives of nonlinear devices. In the cases of IP2 andIP3, up to 2nd and 3rd order derivatives are needed respectively. Thislimits the application of Volterra series in many circuit simulationssince most device models don't provide derivatives higher than firstorder. Also, as the order of Volterra series increases, the complexityof tracking the polynomials grows substantially, and the implementationbecomes more and more complicated.

Thus, there is a need for improved calculation of intermodulationintercept points and other characteristics of circuit distortionanalysis.

SUMMARY OF THE INVENTION

In one embodiment of the present invention, a method of analyzingdistortion in a circuit includes: determining an operating point for thecircuit, where the circuit has a linear offset and a nonlinear offset atthe operating point; determining a first-order solution from the linearoffset and an input having a first input frequency and a second inputfrequency; determining a harmonic component of the nonlinear offset at adifference between the first input frequency and the second inputfrequency; and determining a second-order solution from the linearoffset and the harmonic component of the nonlinear offset at thedifference between the first input frequency and the second inputfrequency. The second-order solution provides an estimate for asecond-order intermodulation product for the circuit.

According to one aspect of this embodiment, the method may furtherinclude determining an estimate for a second-order intercept point forthe circuit from the second-order solution and the first-order solution.Determining the estimate for the second-order intercept point mayfurther include calculating an intersection between a linearrepresentation of the second-order solution and a linear representationof the first-order solution, where the linear representation of thesecond-order solution has a slope of about 2 dB/dB and the linearrepresentation of the first-order solution has a slope of about 1 dB/dB.(Ideally these slopes are exactly 2 dB/dB and 1 dB/dB respectively.)Determining the linear representation of the first-order solution mayfurther include extracting a component from the first-order-solutioncorresponding to one of the first frequency and the second frequency.

According to another aspect, the operating point may be a DC (directcurrent) operating point or a periodic operating point. According toanother aspect, determining the harmonic component of the nonlinearoffset at the difference between the first input frequency and thesecond input frequency may includes: calculating a DFT (Discrete FourierTransform) of the nonlinear offset; and selecting a component of the DFTat the difference between the first input frequency and the second inputfrequency. According to another aspect the circuit may be an analog orRF circuit.

In another embodiment of the present invention, a method of analyzingdistortion in a circuit, includes: determining an operating point forthe circuit, where the circuit has a linear offset and a nonlinearoffset at the operating point; determining a first-order solution fromthe linear offset and an input having a first input frequency and asecond input frequency; determining a harmonic component of thenonlinear offset at a difference between the first input frequency andthe second input frequency; determining a harmonic component of thenonlinear offset at twice the first input frequency; determining asecond-order solution from the linear offset and the harmonic componentof the nonlinear offset at the difference between the first inputfrequency and the second input frequency; determining a second-ordersolution from the linear offset and the harmonic component of thenonlinear offset at twice the first input frequency; determining aharmonic component of the nonlinear offset at a difference between twicethe first input frequency and the second input frequency; anddetermining a third-order solution from the linear offset and theharmonic component of the nonlinear offset at the difference betweentwice the first input-frequency and the second input frequency. Thethird-order solution provides an estimate for a third-orderintermodulation product for the circuit.

This embodiment may include aspects described above. According toanother aspect, the method may further include : determining an estimatefor a third-order intercept point for the circuit from the third-ordersolution and the first-order solution. Determining the estimate for thethird-order intercept point may further include calculating anintersection between a linear representation of the third-order solutionand a linear representation of the first-order solution, where thelinear representation of the third-order solution has a slope of about 3dB/dB and the linear representation of the first-order solution has aslope of about 1 dB/dB. (Ideally these slopes are exactly 3 dB/dB and 1dB/dB respectively.) Determining the linear representation of thefirst-order solution may further include extracting a component from thefirst-order-solution corresponding to one of the first frequency and thesecond frequency.

Additional embodiments relate to an apparatus that includes a computerwith instructions for carrying out any one of the above-describedmethods and a computer-readable medium that stores (e.g., tangiblyembodies) a computer program for carrying out any one of theabove-described methods.

In this way the present invention enables improved calculation ofintermodulation intercept points (e.g., IP2, IP3) and othercharacteristics of circuit distortion analysis (e.g., IM2, IM3).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an exemplary method for calculating second-orderintermodulation products and intercept points according to an embodimentof the present invention.

FIG. 2 shows an exemplary method for calculating third-orderintermodulation products and intercept points according to an embodimentof the present invention.

FIG. 3 shows a diagram of 2nd order intermodulation interactions at RFharmonic ω₁−ω₂.

FIG. 4 shows a diagram of 3rd order intermodulation interaction at RFharmonic 2ω₁−ω₂.

FIG. 5 shows a diagram of 4th order intermodulation interactions at RFharmonic ω₁−ω₂.

FIGS. 6, 7, 8, 9, and 10 show features of simulations related to theembodiments shown in FIGS. 1 and 2.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

1 Introduction

This description presents embodiments of the present invention directedtowards the distortion analysis of circuits. A pertubative approachbased on the Born approximation resolves weakly nonlinear circuit modelswithout requiring explicit high-order device derivatives. Convergenceproperties and the relation to Volterra series are discussed. Accordingto the disclosed methods, IP2 and IP3 can be calculated by 2nd and 3rdorder Born approximations under weakly nonlinear conditions. Adiagrammatic representation of nonlinear interactions is presented.Using this diagrammatic technique, both Volterra series and Bornapproximations can be constructed in a systematic way. The method isgeneralized to calculate other high-order nonlinear effects such as IMn(nth order intermodulation product) and IPn (nth order intermodulationintercept point). In general, the equations are developed in harmonicform and can be implemented in both time and frequency domains foranalog and RF circuits.

First, analytical details for the Born approximation in weakly nonlinearcircuits are discussed. Next methods for calculating IP2 and IP3 arepresented. Next diagrammatic representations are discussed. Nextcomputational results are discussed. In the conclusion additionalembodiments are discussed.

2 The Born Approximation in Weakly Nonlinear Circuits

Consider nonlinear circuit equation $\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}t}{q\left( {V(t)} \right)}} + {i\left( {V(t)} \right)}} = {{{B(t)} +} \in {\cdot {s(t)}}}} & (1)\end{matrix}$where B is the operating point source (e.g., LO or clock signal) and sis the RF input source. Typically, the input s represents a singlefrequency source (e.g., sin(ω₁t)) or a sum of frequency sources (e.g.,sin(ω₁t)+sin(ω₂t)). The parameter ε is introduced to keep track of theorder of perturbation expansion and is set equal to 1 at the end of thecalculations. For simplicity, we use F(v) to denote LHS (left hand side)in equation (1). Assume V₀ is the operating point at zero RF input sothat V₀ satisfies the equationF(V ₀)=B.   (2)

For a LNA (low noise amplifier), V₀ is the DC solution. For a mixer orswitch capacitor filter, V₀ is the periodic steady-state solution underB, and its fundamental frequency is the LO or clock frequency ω_(c).Expanding equation (1) around V₀, we haveF(V ₀)+L·v+F _(NL)(V ₀ ,v)=B+ε·s   (3)where v is the circuit response to RF signal s. Operator L is thelinearization of the LHS in equation (1) and is defined as$\begin{matrix}{{L \cdot v} = {{\frac{\mathbb{d}}{\mathbb{d}t}\left( {\frac{\partial}{\partial V}{{q\left( V_{0} \right)} \cdot v}} \right)} + {\frac{\partial}{\partial V}{{i\left( V_{0} \right)} \cdot {v.}}}}} & (4)\end{matrix}$For periodic V₀(t), L is also periodic and has the same fundamentalfrequency as V₀ at ω_(c). F_(NL) in equation (3) is defined asF _(NL)(V ₀ , v)=F(V ₀ +v)−F(V ₀)−L·v.   (5)That is, F_(NL) is the sum of all nonlinear terms of the circuit.Combining equations (2) and (3), we have the equation for v:L·v+F _(NL)(V ₀ , v)=ε·s.   (6)

In most analog designs, the circuit functions in a nearly linear region.In particular, IP2/IP3 measurements requires very low RF power level soonly the leading order terms are involved. Under such a weakly-nonlinearcondition, the nonlinear term F_(NL) is small compared to the linearterm L·v. We can treat F_(NL) as a perturbation and solve v iteratively:u ^((n)) =v ⁽¹⁾ −L ⁻¹ ·F _(NL)(u ^((n−1))).   (7)Here, u^((n)) is the approximation of v at the nth iteration. And v⁽¹⁾is the small signal solution of order O(ε) obtained from AC (alternatingcurrent) or periodic AC analysis:L·v ⁽¹⁾ =ε·s.   (8)Then in equation (7), u^((n)) is defined for n≧2 where u⁽¹⁾ is definedas v⁽¹⁾. Equation (7) is called the Born approximation or Picarditeration [14, 9, 10]. It is equivalent to successive AC calculationswith F_(NL)(u^((n) ^(—) ¹⁾) being the small signal. As shown in equation(5), the evaluation of F_(NL) is based on the nonlinear device functionF and its first derivative. No higher order derivative is needed. Thisallows us to carry out any order of perturbation without modificationsin current device models. Also, the dynamic range of u^((n)) dependsonly on RF signals. This gives the perturbative approach advantages interms of accuracy.

To estimate the error of u^((n)), subtract equation (6) from equation(7) and linearize around v: $\begin{matrix}{{u^{(n)} - v} = {{{- L^{- 1}} \cdot \left\lbrack {\frac{\partial}{\partial v}{{F_{NL}(v)} \cdot \left( {u^{({n - 1})} - v} \right)}} \right\rbrack} + \ldots}} & (9)\end{matrix}$Recall F_(NL) is a nonlinear function of v. Its lowest order is O(v²).Therefore, we have:∥u^((n))−v∥∝ε·∥u^(n−1))−v∥.   (10)Since ∥v⁽¹⁾−v∥∝O(ε²), we have:∥u^((n))−v∥∝O(ε^(n+1))   (11)and∥u^((n))−u^((n−1))∥∝O(ε^(n)).   (12)Equations (11) and (12) indicate that u^((n)) is accurate to the orderof O(ε^(n)) and the Born approximation converges by O(ε^(n)).

In the situation where LO and RF frequencies are incommensurate witheach other for a particular harmonic Σ_(i)k_(i)·ω_(i)+m·ω_(c) (ω_(i) isthe frequency of an RF signal), its leading order is O(ε^(Σ) ^(i) ^(|k)^(i) ^(|)) and the next leading order is O(ε^(2+Σ) ^(i) ^(|k) ^(i)^(|)). As a result, when n is even, the error in u^((n)) is of orderO(ε^(n+2)) for even harmonics (Σk_(i) is even) and of order O(ε^(n+1))for odd harmonics (Σk_(i) is odd), and vice versa when n is odd. (Thatis, when n is odd the error is of order O(ε^(n+2)) for odd harmonics andof order O(ε^(n+1)) for even harmonics.)

The Born approximation is closely related to Volterra series. Theconnection between them was pointed out by Leon and Schaefer [9] and byLi and Pileggi [10]. In the Volterra series, the nth term v^((n)) is theexact Taylor expansion of v in terms of parameter ε and v^((n)) ∝ε^(n).Because the nth order Born approximation is accurate to O(ε^(n)) asshown in equation (11), u^((n)) is equal to the sum of Volterra seriesup to v^((n)) plus higher order terms:u ^((n)) =v ⁽¹⁾ +v ⁽²⁾ + . . . +v ^((n)) +O(ε^((n+1))).   (13)

Furthermore, for incommensurate frequencies, when n is even, thedifference between the Born approximation and the Volterra series is oforder O(ε^((n+2))) at even harmonics and of order O(ε^(n+1))) at oddharmonics, and vice versa when n is odd. (That is, when n is odd theerror is of order O(ε^(n+2) )for odd harmonics and of order O(ε^(n+1))for even harmonics.)

Equation (7) can be simplified in the form of RF harmonics. Expanding vwith Fourier coefficients v(ω,m) gives: $\begin{matrix}{{v(t)} = {\sum\limits_{\omega}{\sum\limits_{m}{{v\left( {\omega,m} \right)}{\mathbb{e}}^{{j{({\omega + {m\quad\omega_{c}}})}}t}}}}} & (14)\end{matrix}$where ω=Σ_(i)k_(i)·ω_(i) is the RF harmonic frequency. Here we considerthe DC operating point as a special case with ω_(c)=0. We define RFharmonic v_(ω)(t) as $\begin{matrix}{{{v_{\omega}(t)} = {\sum\limits_{m}{{v\left( {\omega,m} \right)}{\mathbb{e}}^{{j{({\omega + {m\quad\omega_{c}}})}}t}}}},} & (15)\end{matrix}$and equation (14) becomes $\begin{matrix}{{v(t)} = {\sum\limits_{\omega}{{v_{\omega}(t)}.}}} & (16)\end{matrix}$It should be emphasized that v₁₀₇ is not the Fourier transform of vbecause it's still a function of time. It's the harmonic component offrequency ω in v. As shown in equation (15), v_(ω) carries frequency ωand all the sidebands of ω_(c). Note that operator L has only ω_(c)harmonics. When frequencies ω_(i) are incommensurate with ω_(c), RFharmonics are orthogonal to each other in equation (7) and u_(ω) can besolved independentlyu _(ω) ^((n)) =v _(ω) ⁽¹⁾ −L ⁻¹ ·F _(NL,ω)(u ^((n−1))).   (17)

To calculate harmonics of F_(NL), we label u_(ω) and F_(NL,ω) with indexvector {right arrow over (k)}=(k₁, k₂, . . . ) asu_({right arrow over (k)}) and _(NL,{right arrow over (k)}). Forincommensurate ω_(i) and ω_(c), {right arrow over (k)} is uniquelydetermined by ω. The following scaling law holds for algebraic functionsq(V) and i(V): $\begin{matrix}{{F_{NL}\left( {\sum\limits_{{\overset{\rightarrow}{k}}^{\prime}}{{\mathbb{e}}^{j\quad{{\overset{\rightarrow}{k}}^{\prime} \cdot \overset{\rightarrow}{\theta}}}{u_{{\overset{\rightarrow}{k}}^{\prime}}(t)}}} \right)} = {\sum\limits_{\overset{\rightarrow}{k}}{{\mathbb{e}}^{j\quad{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{\theta}}} \cdot {{F_{{NL},\overset{\rightarrow}{k}}\left( {u(t)} \right)}.}}}} & (18)\end{matrix}$Equation (18) demonstrates that if we multiply everyu_({right arrow over (k)}) with a factore^(j{right arrow over (k)}·{right arrow over (θ)}) and compute F_(NL),then F_(NL,{right arrow over (k)}) is the Fourier transform of F_(NL) asa function of {right arrow over (θ)}. Using multidimensional DFT(Discrete Fourier Transform), we calculate F_(NL,{right arrow over (k)})as $\begin{matrix}{{F_{{NL},\overset{\rightarrow}{k}}\left( {u(t)} \right)} = {\sum\limits_{\overset{\rightarrow}{\theta}}{{\Gamma\left( {\overset{\rightarrow}{k},\overset{\rightarrow}{\theta}} \right)} \cdot {{F_{NL}\left( {\sum\limits_{{\overset{\rightarrow}{k}}^{\prime}}{{\mathbb{e}}^{j\quad{\overset{\rightarrow}{k} \cdot \overset{\rightarrow}{\theta}}}{u_{\overset{\rightarrow}{k}}(t)}}} \right)}.}}}} & (19)\end{matrix}$where Γ({right arrow over (k)},{right arrow over (θ)}) is the DFT matrixand {right arrow over (θ)} is uniformly distributed in (−π, π) for eachθ_(i). Equation (19) has also been used in harmonic balance forfrequency domain function evaluation [3]. It is equivalent tocalculating derivatives of F(V) numerically. Therefore, althoughhigh-order device derivatives are absent from equation (17), accuratenonlinear calculations still require smooth I/V or Q/V characteristicsover the range of voltages or currents in which the circuit operates.

Equation (17) can be implemented in both time and frequency domains[15]. In a frequency representation, u_(ω) and F_(NL,ω) are expandedwith ω_(c) harmonics as in equation (15).

3 IP2 and IP3 Calculations

In the 2nd order Born approximation, u_(ω) ₁ _(−ω) ₂ ⁽²⁾ is accurate tothe leading order O(ε²). The next-order terms are of order O(ε⁴), andthey are negligible compared to O(ε²) terms when RF input signals areweak. Therefore, we can use u_(ω) ₁ _(−ω) ₂ ⁽²⁾ at a low RF power levelas an estimate for IM2 to calculate IP2. We can solve for u_(ω) ₁ _(−ω)₂ ⁽²⁾ from the linear solution u^((1):)u _(ω) ₁ _(−ω) ₂ ⁽²⁾ =−L ⁻¹ ·F _(NL,ω) ₁ _(−ω) ₂ (u ⁽¹⁾)   (20)This computation takes one DC or periodic steady-state calculation atzero RF input and three AC or periodic AC solutions (two for u⁽¹⁾ andone for u⁽²⁾). Unlike multi-tone simulations, where the size of theJacobian matrix is proportional to the number of harmonics used, thesize of L in the perturbation is independent of harmonic number, whichaffects only function evaluation in equation (19).

This process is summarized in FIG. 1. A method 102 for calculatingIM2/IP2 begins with determining 104 an operating point for the circuit.The operating point can be a single point (e.g., a DC operating point)or an interval (e.g., a periodic operating point). Once an operatingpoint is set, a linear offset (e.g., L in eq. (4)) and a nonlinearoffset (e.g., F_(NL) in eq. (5)) can be determined for perturbationanalysis about the operating point. Next, a first-order solution (e.g.,u⁽¹⁾=v⁽¹⁾ in eq. (8)) can be determined 106 from the linear offset andan input (e.g., s in eq. (8)) having a first input frequency (e.g., ω₁)and a second input frequency (e.g., ω₂). For example, we can takes=s₁+s₂ where s₁=sin(ω₁t) and s₂=sin(ω₂t), and where the solutions canbe obtained separately for each frequency and summed because of thelinearity of equation (8). These two frequencies are typically close invalue (e.g., ω₁≈ω₂).

Next a harmonic component of the nonlinear offset can be determined 108at a difference between the first input frequency and the second inputfrequency (e.g., F_(NL) at ω₁−ω₂ in eq. (20)). Next a second-ordersolution (e.g., u⁽²⁾ at ω₁−ω₂ in eq. (20)) can be determined 110 fromthe linear offset and the harmonic component of the nonlinear offset atthe difference between the first input frequency and the second inputfrequency. As discussed above, this second-order solution provides anestimate for the second-order intermodulation product (IM2) for thecircuit.

The method 102 can be extended by determining 112 an estimate for asecond-order intercept point (e.g., IP2) for the circuit from thesecond-order solution and the first-order solution. Typically, thisincludes calculating an intersection on a log-log scale between a linearrepresentation of the second-order solution and a linear representationof the first-order solution. Ideally, the linear representation of thesecond-order solution has a slope of 2 dB/dB and the linearrepresentation of the first-order solution has a slope of 1 dB/dB.Typically, only one frequency is used for the characterization of thefirst-order solution so that a component corresponding to either thefirst frequency (e.g., ω₁) or the second frequency (e.g., ω₂) isextracted from the first-order solution. Details for calculatingintermodulation intercept points from intermodulation products (e.g,calculating IP2 from IM2 and IP3 from IM3) are well-known in the art ofcircuit analysis (e.g., [8] at p. 298, [13] at p. 18).

Similarly, for the IP3 calculation we can use the 3rd order Bornapproximation u_(2ω) ₁ _(−ω) ₂ ⁽³⁾ at low power level as an estimate forIM3. It is accurate to the leading order O(ε³) and next order terms areof O(ε⁵). We solve for u_(2ω1−ω2) ⁽³⁾ from the 2nd order solution u⁽²⁾:u _(2ω) ₁ _(−ω) ₂ ⁽³⁾ =−L ⁻¹ ·F _(NL,2ω) ₁ _(−ω) ₂ (u ⁽²⁾).   (21)Note that the ε³ terms in u_(2ω) ₁ _(−ω) ₂ ⁽³⁾ arise from second andthird order polynomials in F_(NL). Only a handful of RF harmonics inu⁽²⁾ are involved in F_(NL,2ω) ₁ _(−ω) ₂ at order O(ε³). We demonstratethis by expanding F_(NL) in equation (21) with polynomials:$\begin{matrix}{u_{{2\omega_{1}} - \omega_{2}}^{(3)} = {{- L^{- 1}} \cdot {\begin{Bmatrix}{{F_{2} \cdot \left\lbrack {{u_{- \omega_{2}}^{(1)} \cdot u_{2\omega_{1}}^{(2)}} + {u_{\omega_{1}}^{(1)} \cdot u_{\omega_{1} - \omega_{2}}^{(2)}}} \right\rbrack} +} \\{{\frac{1}{2}{F_{3} \cdot u_{\omega_{1}}^{(1)} \cdot u_{\omega_{1}}^{(1)} \cdot u_{- \omega_{2}}^{(1)}}} + {O\left( ɛ^{5} \right)}}\end{Bmatrix}.}}} & (22)\end{matrix}$

In equation (22), the linear operator F_(n) is the nth derivative of Fwith respect to V at V₀ and is given by $\begin{matrix}{F_{n} = {{\frac{\mathbb{d}}{\mathbb{d}t}\left( {\frac{\partial^{n}}{\partial V^{n}}{q\left( V_{0} \right)}} \right)} + {\frac{\partial^{n}}{\partial V^{n}}{{q\left( V_{0} \right)} \cdot \frac{\mathbb{d}}{\mathbb{d}t}}} + {\frac{\partial^{n}}{\partial V^{n}}{{i\left( V_{0} \right)}.}}}} & (23)\end{matrix}$Without losing accuracy we can write equation (22) as $\begin{matrix}{u_{{2\omega_{1}} - \omega_{2}}^{(3)} = {{{- L^{- 1}} \cdot {F_{{NL},{{2\omega_{1}} - \omega_{2}}}\left\lbrack {u_{\omega_{1}}^{(1)} + u_{- \omega_{2}}^{(1)} + u_{2\omega_{1}}^{(2)} + u_{\omega_{1} - \omega_{2}}^{(2)}} \right\rbrack}} + {{O\left( ɛ^{5} \right)}.}}} & (24)\end{matrix}$

Equation (24) shows that to solve u_(2ω) ₁ _(−ω) ₂ ⁽³⁾ to O(ε³)accuracy, we only need u_(2ω) ₁ ⁽²⁾ and u_(ω) ₁ _(−ω) ₂ ⁽²⁾ from thesecond order approximation, for which we haveu _(2ω) ₁ ⁽²⁾ =−L ⁻¹ ·F _(NL,2ω) ₁ (u ⁽¹⁾)   (25)andu _(ω) ₁ _(−ω) ₂ ⁽²⁾ =−L ⁻¹ ·F _(NL,ω) ₁ _(−ω) ₂ (u ⁽¹⁾).   (26)Other harmonics contribute to higher order terms, and therefore can beignored. The computation takes one DC or periodic steady-statecalculation at zero RF input and five AC or periodic AC solutions (twofor u⁽¹⁾, two for u⁽²⁾, and one for u⁽³⁾). Note that equation (26) isidentical to equation (20) so that an estimate for IM2 is alsoavailable.

This process is summarized in FIG. 2, and the details are similar to thedescription of FIG. 1 above. A method 202 for calculating IM3/IP3 beginswith determining 204 an operating point (e.g., DC or periodic) for thecircuit. Once an operating point is set, a linear offset (e.g., L in eq.(4)) and a nonlinear offset (e.g., F_(NL) in eq. (5)) can be determinedfor perturbation analysis about the operating point. Next, a first-ordersolution (e.g., u⁽¹⁾=v⁽¹⁾ in eq. (8)) can be determined 206 from thelinear offset and an input (e.g., s in eq. (8)) having a first inputfrequency (e.g., ω₁) and a second input frequency (e.g., ω₂).

Next a harmonic component of the nonlinear offset can be determined 208at a difference between the first input frequency and the second inputfrequency (e.g., F_(NL) at ω₁−ω₂ in eq. (26)), and a harmonic componentof the nonlinear offset can be determined 208 at twice the first inputfrequency (e.g., F_(NL) at 2ω₁ in eq. (25)). Next a second-ordersolution (e.g., u⁽²⁾ at ω₁−ω₂ in eq. (26)) can be determined 210 fromthe linear offset and the harmonic component of the nonlinear offset atthe difference between the first input frequency and the second inputfrequency, and a second-order solution (e.g., u⁽²⁾ at 2ω₁ in eq. (25))can be determined 210 from the linear offset and the harmonic componentof the nonlinear offset at twice the first input frequency. Next, aharmonic component of the nonlinear offset (e.g., F_(NL) at 2ω₁−ω₂ ineq. (24)), can be determined 212 at a difference between twice the firstinput frequency and the second input frequency. Next a third-ordersolution (e.g., u⁽³⁾ at 2ω₁−ω₂ in eq. (24)) can be determined 214 fromthe linear offset and the harmonic component of the nonlinear offset atthe difference between twice the first input frequency and the secondinput frequency. As discussed above, this third-order solution providesan estimate for the third-order intermodulation product (IM3) for thecircuit.

The method 202 can be extended by determining 216 an estimate for athird-order intercept point (e.g., IP3) for the circuit from thesecond-order solution and the first-order solution. Typically, thisincludes calculating an intersection on a log-log scale between a linearrepresentation of the third-order solution and a linear representationof the first-order solution. Ideally, the linear representation of thethird-order solution has a slope of 3 dB/dB and the linearrepresentation of the first-order solution has a slope of 1 dB/dB.Details are standard and similar to the above discussion with respect toFIG. 1.

As a heuristic, one can compare the Born approximation with the Volterraseries, where IM2 is calculated byv _(ω) ₁ _(−ω) ₂ ⁽²⁾ =−L ⁻¹ ·[F ₂ ·v _(ω) ₁ ⁽¹⁾ ·v _(−ω) ₂ ⁽¹⁾]  (27)and IM3 is calculated by $\begin{matrix}{v_{{2\omega_{1}} - \omega_{2}}^{(3)} = {{- L^{- 1}} \cdot \left\{ {{F_{2} \cdot \left\lbrack {{v_{- \omega_{2}}^{(1)} \cdot v_{2\omega_{1}}^{(2)}} + {v_{\omega_{1}}^{(1)} \cdot v_{\omega_{1} - \omega_{2}}^{(2)}}} \right\rbrack} + {\frac{1}{2}{F_{3} \cdot v_{\omega_{1}}^{(1)} \cdot v_{\omega_{1}}^{(1)} \cdot v_{- \omega_{2}}^{(1)}}}} \right\}}} & (28) \\{{v_{2\omega_{1}}^{(2)} = {{- L^{- 1}} \cdot \left\lbrack {\frac{1}{2!}{F_{2} \cdot v_{\omega_{1}}^{(1)} \cdot v_{\omega_{1}}^{(1)}}} \right\rbrack}}{and}} & (29) \\{v_{\omega_{1} - \omega_{2}}^{(2)} = {{- L^{- 1}} \cdot {\left\lbrack {F_{2} \cdot v_{\omega_{1}}^{(1)} \cdot v_{- \omega_{2}}^{(1)}} \right\rbrack.}}} & (30)\end{matrix}$

Equations (20) and (24)-(26) can be considered as approximation ofequations (27)-(30) by replacing all the v^((n)) with u^((n)) and thepolynomials with F_(NL). The actual polynomial multiplication happensimplicitly when evaluating nonlinear function F_(NL) in the Bornapproximation. The deviation introduced is of order O(ε⁴) for IM2 andO(ε⁵) for IM3. Its effects on IP2/IP3 can be estimated by checking thescaling of IM2/IM3 results at different RF power levels. Compared toequations (27)-(30), not only are equations (20) and (24)-(26) free ofF₂ and F₃, but they are also much simpler to implement.

4 Diagrammatic Representation

Equations (27)-(30) are leading order terms in RF harmonics ω₁−ω₂ and2ω₁−ω₂. We illustrate them in FIG. 3 and FIG. 4 using diagrams analogousto the well-known Feynman diagrams in quantum field theory [6].

In FIGS. 3 and 4, each vertex represents a polynomial in F_(NL). Avertex has one outgoing line and at least two incoming lines. Linescoming into the vertex are lower order solutions involved in thepolynomial. The outgoing line is the solution at the current order. TheRF frequency at the outgoing line is equal to the sum of the RFfrequencies at the incoming lines. The three diagrams in FIG. 4correspond to three nonlinear terms in the RHS (right-hand side) ofequation (28). Using the diagrammatic technique, it's convenient toidentify intermediate harmonics we need-to solve in lower-orderperturbations.

In general, to count all diagrams for a given RF frequency at nth order,one can start with the diagram for the v^(n) term of the Taylorexpansion about the operating point, which is the highest-orderpolynomial that needs to be considered. All its incoming lines are offirst order. With the same outgoing frequency, diagrams of v^(n−1) termscan be constructed by merging a pair of incoming lines in the v^(n)diagram into a single line. The frequency at the new line is equal tothe sum of frequencies at the two merged lines. Similarly, diagrams ofother polynomials are obtained by merging more lines. Note that theorder of a new line is the sum of orders of lines being merged.

As an example, we draw diagrams of intermodulation ω₁−ω₂ at 4th order inFIG. 5. All lines are of leading order except for two F₂ diagrams thathave incoming line v_(ω) ₁ ⁽³⁾ or v_(−ω) ₂ ⁽³⁾. (See FIG. 5, third row,first and fourth diagrams from the left.) They are identical to the 2ndorder diagram shown in FIG. 3 except that the incoming lines in FIG. 3are v_(ω) ₁ ⁽¹⁾ and v_(−ω) ₂ ⁽¹⁾. To distinguish them from the 2nd orderdiagram, ω₁ or −ω₂ is explicitly marked as 3rd order. FIG. 5 shows thatthe 4th order calculation of ω₁−ω₂ takes harmonics 2ω₁−ω₂, ω₁−2ω₂, ω₁and −ω₂ from the 3rd order solution and harmonics ω₁−ω₂, ω₁+ω₂, −ω₁−ω₂,2ω₁, −2ω₁ and 0 from the 2nd order. Employing the technique recursivelyto lower-order lines, we obtain the list of harmonics need to be solvedat each order of the perturbation. The error term in final solution isof order O(ε⁶).

In addition, We define the following rules to carry out perturbationcalculations directly from the diagrams. For Volterra series we have:

1. To each incoming external line, assign v_(ω) ^((n)) where ω is the RFfrequency of the line.

2. To each internal line, assign operator L⁻ ¹ , which represents theGreen's function or propagator of linear equation (8). For a line atfrequency ω, L⁻ ¹ has the form of G_(ω)(t, t′) in time domain andG(ω+mω_(c), ω+m′ω_(c)) in frequency domain.

3. To each vertex, assign operator −F_(n)/n! and operate it on theproduct of incoming lines with a permutation factor to compute thecorresponding polynomial.

4. To the outgoing external line, assign L⁻ ¹ for the final AC orperiodic AC calculation.

These rules are analogous to the Feynman rules [6]. Applying these rulesfor the Volterra series (or the Volterra approximation) to the diagramsin FIGS. 3 and 4 results in: $\begin{matrix}{{v_{\omega_{1} - \omega_{2}}^{(2)} = {{- \frac{1}{L}} \cdot F_{2} \cdot v_{\omega_{1}}^{(1)} \cdot v_{- \omega_{2}}^{(1)}}}{and}} & (31) \\{v_{{2\omega_{1}} - \omega_{2}}^{(3)} = {{\frac{1}{L} \cdot F_{2} \cdot v_{- \omega_{2}}^{(1)} \cdot \left( {{\frac{1}{L} \cdot \frac{1}{2!}}{F_{2} \cdot v_{\omega_{1}}^{(1)} \cdot v_{\omega_{1}}^{(1)}}} \right)} + {\frac{1}{L} \cdot F_{2} \cdot v_{\omega_{1}}^{(1)} \cdot \left( {\frac{1}{L} \cdot F_{2} \cdot v_{\omega_{1}}^{(1)} \cdot v_{- \omega_{2}}^{(1)}} \right)} - {\frac{1}{L} \cdot \frac{1}{2} \cdot F_{3} \cdot v_{\omega_{1}}^{(1)} \cdot v_{\omega_{1}}^{(1)} \cdot {v_{- \omega_{2}}^{(1)}.}}}} & (32)\end{matrix}$Equations (31) and (32) are equivalent to equations (27)-(30). Replacingv⁽¹⁾ with L⁻ ¹ ·s in equation (31) and equation (32) yields the 2nd and3rd order Volterra kernels for ω₁−ω₂ and 2ω₁−ω₂ respectively.

Similarly, for the Born approximation we have:

1. To each incoming external line, assign u_(ω) ₁ ^((n)) where ω is theRF frequency of the line.

2. To each internal line, assign operator −L⁻¹

3. To each vertex, assign nonlinear function −F_(NL,ω)(·), where ω isthe RF frequency of the outgoing line of the vertex. The input variableto the function −F_(NL,ω)(·) is the sum of incoming lines with distinctfrequencies. If a vertex has multiple incoming lines at an identicalfrequency, only the one with highest order is included in the input to−F_(NL,ω(·)) because it already contains all previous orders.

4. To the outgoing external line, assign −L⁻¹ for the final AC or periodAC calculation.

Applying these rules for the Born approximation to the diagrams in FIGS.3 and 4 results in: $\begin{matrix}{{u_{\omega_{1} - \omega_{2}}^{(2)} = {{- \frac{1}{L}}{{F_{{NL},{\omega_{1} - \omega_{2}}}( \cdot )}\left\lbrack {u_{\omega_{1}}^{(1)} + u_{- \omega_{2}}^{(1)}} \right\rbrack}}}{and}} & (33) \\{u_{{2\omega_{1}} - \omega_{2}}^{(3)} = {{{- \frac{1}{L}}{{F_{{NL},{{2\omega_{1}} - \omega_{2}}}( \cdot )}\left\lbrack {u_{- \omega_{2}}^{(1)} - {\frac{1}{L}{F_{{NL},{2\omega_{1}}}( \cdot )}u_{\omega_{1}}^{(1)}}} \right\rbrack}} - {\frac{1}{L}{{F_{{NL},{{2\omega_{1}} - \omega_{2}}}( \cdot )}\left\lbrack {u_{\omega_{1}}^{(1)} - {\frac{1}{L}{F_{{NL},{\omega_{1} - \omega_{2}}}( \cdot )}\left( {u_{\omega_{1}}^{(1)} + u_{- \omega_{2}}^{(1)}} \right)}} \right\rbrack}} - {\frac{1}{L}{{{F_{{NL},{{2\omega_{1}} - \omega_{2}}}( \cdot )}\left\lbrack {u_{\omega_{1}}^{(1)} + u_{- \omega_{2}}^{(1)}} \right\rbrack}.}}}} & (34)\end{matrix}$

Note that in the first and third terms of equation (34) we use one u_(ω)₁ ⁽¹⁾ to represent the two ω₁ incoming lines at vertices F_(NL,2ω) ₂ andF_(NL,2ω) ₁ _(−ω) ₂ . Equations (24)-(26) can be obtained from equation(34) by combining the three terms with F_(NL,2ω) ₂ _(−ω) ₂ (·) into oneand eliminating identical terms in equation (34). (FIGS. 3 and 4correspond to FIGS. 1 and 2 respectively.) Using the diagrammaticmethod, we can systematically construct Born approximations or Volterraseries for calculating (or representing) any high-order perturbation.That is, the rules described above can be implemented as methods forcalculating perturbations to any order using the Born approximation orthe Volterra series. Alternatively, perturbations can be calculated athigher orders by automated software for symbolic manipulation ofpolynomial and multinomial functions.

5 Computational Results

Tests were performed on five circuits including an LNA and a mixer. FIG.6 shows basic circuit information (e.g., number of active and passiveelements and number of nodes), and FIG. 7 shows the relevant operatingparameters (i.e., RF frequencies ω₁, ω₂ and LO frequencies ω_(c)).

Intermodulation products were calculated using the Born approximation infrequency domain (e.g, as in FIGS. 1 and 2), and the results (orrepresentations) were compared with 2-tone (for the LNA) and 3-tone (forthe mixer) harmonic balance (HB) simulations [3]. FIG. 8 shows thecalculated IM2 at two different RF power levels in Mixer1 and Mixer2 at0.88 MHz and 0.2 MHz, respectively (i.e., as values for ω₁−ω₂). FIG. 9shows the calculated IM3 in LNA, Mixer2, Mixer3 and Mixer4 at 2.452 GHz,2.2 MHz, 30 MHz and 80 KHz, respectively (i.e., as values for 2ω₁−ω₂ forthe LNA and for 2ω₁−ω₂−ω_(c) for the mixers). (Here the Bornapproximation is denoted as a “perturbation” in the figures. Note thatthe ω₁-component of the input was used to determine the RF power.) Theseperturbation results are in excellent agreement with multi-tonesimulations. It is worthwhile to point out that standard device modelssuch as BSIM3 and BJT are used in the simulations. As noted above,Volterra series calculations are generally not possible withoutmodifying the models to compute 2nd and 3rd order derivatives.

To verify the scaling behavior of IM2/IM43 at different RF signallevels, we computed the slope of IM2/IM3 power as a function of RF inputpower using the results shown in FIGS. 8 and 9. The IM2 slopes wereestimated to be 2.00006 dB/dB in Mixer1 and 2.0083 dB/dB in Mixer2. TheIM3 slopes in LNA, Mixer2, Mixer3 and Mixer4 were estimated to be 3.0039dB/dB, 2.9976 dB/dB, 3.0097 dB/dB and 2.99998 dB/dB, respectively. Theseresults, which are very close to the theoretical values of 2 dB/dB and 3dB/dB, demonstrate that at the specified power level nonlineardistortion is dominated by leading order effects and higher ordercontributions are insignificant in these circuits. Therefore, IP2 andIP3 can be accurately extracted from the calculations.

FIG. 10 shows a speed comparison between the Bom approximation andmulti-tone simulation. Here the computation time used to solve DC orperiodic operating point was included for the Bom approximation (i.e.,perturbation method). According to these results, performance isimproved by three to six time using the perturbative method.

6 Conclusion

As demonstrated above by embodiments of the present invention, IP2 andIP3 can be calculated by using 2nd and 3rd order Bom approximationsunder weakly nonlinear condition. The approach does not requirehigh-order device derivatives and, in general, can be implementedwithout modification of device models. The approach is formulated assuccessive small signal calculations. Since RF signals are treated asperturbation to the operating point, the dynamic range only needs tocover RF excitations. In general, the computation takes one DC orperiodic steady-state calculation at zero RF input and three (for IP2)or five (for IP3) AC or periodic AC solutions, regardless of the numberof harmonics used. As demonstrated by the calculation shown above, thisapproach gives results that are consistent with the conventionalmulti-tone simulation methods for calculating IP2 and IP3. However thisapproach is typically much more efficient compared to the conventionalsimulations.

A diagrammatic representation was introduced to analyze nonlinearinteractions at a given order. Relevant intermediate harmonics inlower-order approximations are identified using this diagrammatictechnique. Perturbations can be constructed directly from diagrams in asystematic way. The approach can be applied to calculating otherhigh-order distortions such as arbitrary intermodulation products (IMn),and these distortions can be used to calculate correspondingintermodulation intercept points (IPn)

Additional embodiments relate to an apparatus that includes a computerwith instructions for carrying out any one of the above-describedmethods. In this context the computer may be a general-purpose computerincluding, for example, a central processing unit, memory, storage andInput/Output devices. However, the computer may include a specializedmicroprocessor or other hardware for carrying out some or all aspects ofthe methods. Additional embodiments also include a computer-readablemedium that stores (e.g., tangibly embodies) a computer program forcarrying out any one of the above-described methods by means of acomputer.

Although only certain exemplary embodiments of this invention have beendescribed in detail above, those skilled in the art will readilyappreciate that many modifications are possible in the exemplaryembodiments without materially departing from the novel teachings andadvantages of this invention. For example, aspects of embodimentsdisclosed above can be combined in other combinations to form additionalembodiments. Accordingly, all such modifications are intended to beincluded within the scope of this invention.

7 References

The following references are related to the disclosed subject matter:

-   [1] J. Bussgang, L. Ehrman, and J. Graham. Analysis of nonlinear    systems with multiple inputs. Proc. IEEE, 62:1088-1119, August 1974.-   [2] S. Chisholm and L. Nagel. Efficient computer simulation of    distortion in electronic circuits. IEEE Trans. on Circuits and    Systems, 20(6):742-745, November 1973.-   [3] P. Feldmann, B. Melville, and D. Long. Efficient frequency    domain analysis of large nonlinear analog circuits. Proc. Custom    Integrated Circuits Conf., pages 461-464, 1996.-   [4] D. Feng, J. Phillips, K. Nabors, K. Kundert, and J. White.    Efficient computation of quasi-periodic circuit operating conditions    via a mixed frequency/time approach. Proc. of 36th DAC pages    635-640, June 99.-   [5] J. Haywood and Y. L. Chow. Intermodulation distortion analysis    using a frequency-domain harmonic balance technique. IEEE Trans. on    Microwave Theory and Techniques, 36(8):1251-1257, August 1988.-   [6] C. Itzykson and J. Zuber. Quantum Field Theory. McGraw-Hill    Inc., 1980.-   [7] Y. Kuo. Distortion analysis of bipolar transistor circuits. IEEE    Trans. on Circuits and Systems, 20(6):709-716, November 1973.-   [8] T. Lee. The Design of CMOS Radio-Frequency Integrated Circuits.    Cambridge University Press, Cambridge, UK, 1998.-   [9] B. Leon and D. Schaefer. Volterra series and picard iteration    for nonlinear circuits and systems. IEEE Trans. on Circuits and    Systems, 25(9):789-793, September 1978.-   [10] P. Li and L. Pileggi. Efficient per-nonlinearity distortion    analysis for analog and rf circuits. IEEE Trans. on Computer-Aided    Design of Integrated Circuits and Systems, 22(10):1297-1309, October    2003.-   [11] S. Maas. Nonlinear Microwave Circuits. Artech House, Norwood,    Mass., 1988.-   [12] S. Narayanan. Application of volterra series to intermodulation    distortion analysis of transistor feedback amplifiers. IEEE Trans.    on Circuits and Systems, 17(4):518-527, November 1970.-   [13] B. Razavi. RF Microelectronics. Prentice Hall, Upper Saddle    River, N.J., 1998.-   [14] J. Sakurai. Modern Quantum Mechanics. The Benjamin/Cummings    Publishing Company, Inc., Menlo Park, Calif., 1985.-   [15] R. Telichevesky, K. Kundert, and J. White. Efficient ac and    noise analysis of two-tone rf circuits. Proc. of 33th DAC pages    292-297, June 96.-   [16] P. Wambacq, G. Gielen, P. Kinget, and W. Sansen. High-frequency    distortion analysis of analog integrated circuits. IEEE Trans. on    Circuits and Systems II, 46(3):335-345, March 1999.-   [17] F. Yuan and A. Opal. Distortion analysis of periodically    switched nonlinear circuits using time-varying volterra series. IEEE    Trans. on Circuits and Systems I, 48(6):726-738, June 2001.

1. A method of analyzing distortion in a circuit, comprising:determining an operating point for the circuit, the circuit having alinear offset and a nonlinear offset at the operating point; determininga first-order solution from the linear offset and an input having afirst input frequency and a second input frequency; determining aharmonic component of the nonlinear offset at a difference between thefirst input frequency and the second input frequency; and determining asecond-order solution from the linear offset and the harmonic componentof the nonlinear offset at the difference between the first inputfrequency and the second input frequency, wherein the second-ordersolution provides an estimate for a second-order intermodulation productfor the circuit.
 2. A method according to claim 1, further comprising:determining an estimate for a second-order intercept point for thecircuit from the second-order solution and the first-order solution. 3.A method according to claim 2, wherein determining the estimate for thesecond-order intercept point includes: calculating an intersectionbetween a linear representation of the second-order solution and alinear representation of the first-order solution, wherein the linearrepresentation of the second-order solution has a slope of about 2 dB/dBand the linear representation of the first-order solution has a slope ofabout 1 dB/dB.
 4. A method according to claim 3, wherein determining thelinear representation of the first-order solution includes extracting acomponent from the first-order-solution corresponding to one of thefirst frequency and the second frequency.
 5. A method according to claim1, wherein the operating point is a DC (direct current) operating point.6. A method according to claim 1, wherein the operating point is aperiodic operating point.
 7. A method according to claim 1, furthercomprising: calculating a DFT (Discrete Fourier Transform) of thenonlinear offset; wherein determining the harmonic component of thenonlinear offset at the difference between the first input frequency andthe second input frequency includes selecting a component of the DFT atthe difference between the first input frequency and the second inputfrequency.
 8. A method according to claim 1, wherein the circuit is ananalog or RF (radio frequency) circuit.
 9. A method of analyzingdistortion in a circuit, comprising: determining an operating point forthe circuit, the circuit having a linear offset and a nonlinear offsetat the operating point; determining a first-order solution from thelinear offset and an input having a first input frequency and a secondinput frequency; determining a harmonic component of the nonlinearoffset at a difference between the first input frequency and the secondinput frequency; determining a harmonic component of the nonlinearoffset at twice the first input frequency; determining a second-ordersolution from the linear offset and the harmonic component of thenonlinear offset at the difference between the first input frequency andthe second input frequency; determining a second-order solution from thelinear offset and the harmonic component of the nonlinear offset attwice the first input frequency; determining a harmonic component of thenonlinear offset at a difference between twice the first input frequencyand the second input frequency; and determining a third-order solutionfrom the linear offset and the harmonic component of the nonlinearoffset at the difference between twice the first input frequency and thesecond input frequency, wherein the third-order solution provides anestimate for a third-order intermodulation product for the circuit. 10.A method according to claim 9, further comprising: determining anestimate for a third-order intercept point for the circuit from thethird-order solution and the and the first-order solution.
 11. A methodaccording to claim 10, wherein determining the estimate for thethird-order intercept point includes: calculating an intersectionbetween a linear representation of the third-order solution and a linearrepresentation of the first-order solution, wherein the linearrepresentation of the third-order solution has a slope of about 3 dB/dBand the linear representation of the first-order solution has a slope ofabout 1 dB/dB.
 12. A method according to claim 11, wherein determiningthe linear representation of the first-order solution includesextracting a component from the first-order solution corresponding toone of the first frequency and the second frequency.
 13. A methodaccording to claim 9, further comprising: calculating a DFT of thenonlinear offset; wherein determining the harmonic component of thenonlinear offset at the difference between the first input frequency andthe second input frequency includes selecting a component of the DFT atthe difference between the first input frequency and the second inputfrequency, determining the harmonic component of the nonlinear offset attwice the first input frequency includes selecting a component of theDFT at twice the first input frequency; and determining the harmoniccomponent of the nonlinear offset at the difference between twice thefirst input frequency and the second input frequency includes selectinga component of the DFT at the difference between twice the first inputfrequency and the second input frequency.
 14. A method of analyzingdistortion in a circuit, comprising: a step for representing anoperating point for the circuit, the circuit having a linear offset anda nonlinear offset at the operating point; a step for representing afirst-order solution from the linear offset and an input having a firstinput frequency and a second input frequency; a step for representing aharmonic component of the nonlinear offset at a difference between thefirst input frequency and the second input frequency; and a step forrepresenting a second-order solution from the linear offset and theharmonic component of the nonlinear offset at the difference between thefirst input frequency and the second input frequency, wherein thesecond-order solution provides an estimate for a second-orderintermodulation product for the circuit.
 15. A method according to claim14, further comprising: a step for representing a second-order interceptpoint for the circuit from the second-order solution and the first-ordersolution.
 16. A method according to claim 14, further comprising: a stepfor representing a harmonic component of the nonlinear offset at twicethe first input frequency; a step for representing a second-ordersolution from the linear offset and the harmonic component of thenonlinear offset at twice the first input frequency; a step forrepresenting a harmonic component of the nonlinear offset at adifference between twice the first input frequency and the second inputfrequency; and a step for representing a third-order solution from thelinear offset and the harmonic component of the nonlinear offset at thedifference between twice the first input frequency and the second inputfrequency, wherein the third-order solution provides an estimate for athird-order intermodulation product for the circuit.
 17. A methodaccording to claim 16, further comprising: a step for representing athird-order intercept point for the circuit from the third-ordersolution and the and the first-order solution.
 18. An apparatus foranalyzing distortion in a circuit, the apparatus comprising a computerfor executing computer instructions, wherein the computer includescomputer instructions for: determining an operating point for thecircuit, the circuit having a linear offset and a nonlinear offset atthe operating point; determining a first-order solution from the linearoffset and an input having a first input frequency and a second inputfrequency; determining a harmonic component of the nonlinear offset at adifference between the first input frequency and the second inputfrequency; and determining a second-order solution from the linearoffset and the harmonic component of the nonlinear offset at thedifference between the first input frequency and the second inputfrequency, wherein the second-order solution provides an estimate for asecond-order intermodulation product for the circuit.
 19. An apparatusaccording to claim 18, wherein the computer further includes computerinstructions for: determining an estimate for a second-order interceptpoint for the circuit from the second-order solution and the first-ordersolution.
 20. An apparatus according to claim 19, wherein determiningthe estimate for the second-order intercept point includes: calculatingan intersection between a linear representation of the second-ordersolution and a linear representation of the first-order solution,wherein the linear representation of the second-order solution has aslope of about 2 dB/dB and the linear representation of the first-ordersolution has a slope of about 1 dB/dB.
 21. An apparatus according toclaim 18, wherein the computer further includes computer instructionsfor calculating a DFT of the nonlinear offset, and determining theharmonic component of the nonlinear offset at the difference between thefirst input frequency and the second input frequency includes selectinga component of the DFT at the difference between the first inputfrequency and the second input frequency.
 22. An apparatus according toclaim 18, wherein the computer further includes computer instructionsfor determining a harmonic component of the nonlinear offset at twicethe first input frequency; determining a second-order solution from thelinear offset and the harmonic component of the nonlinear offset attwice the first input frequency; determining a harmonic component of thenonlinear offset at a difference between twice the first input frequencyand the second input frequency; and determining a third-order solutionfrom the linear offset and the harmonic component of the nonlinearoffset at the difference between twice the first input frequency and thesecond input frequency, wherein the third-order solution provides anestimate for a third-order intermodulation product for the circuit. 23.An apparatus according to claim 22, wherein the computer furtherincludes computer instructions for: determining an estimate for athird-order intercept point for the circuit from the third-ordersolution and the and the first-order solution.
 24. An apparatusaccording to claim 23, wherein determining the estimate for thethird-order intercept point includes: calculating an intersectionbetween a linear representation of the third-order solution and a linearrepresentation of the first-order solution, wherein the linearrepresentation of the third-order solution has a slope of about 3 dB/dBand the linear representation of the first-order solution has a slope ofabout 1 dB/dB.
 25. An apparatus according to claim 22, wherein thecomputer further includes computer instructions for calculating a DFT ofthe nonlinear offset, determining the harmonic component of thenonlinear offset at the difference between the first input frequency andthe second input frequency includes selecting a component of the DFT atthe difference between the first input frequency and the second inputfrequency, determining the harmonic component of the nonlinear offset attwice the first input frequency includes selecting a component of theDFT at twice the first input frequency, and determining the harmoniccomponent of the nonlinear offset at the difference between twice thefirst input frequency and the second input frequency includes selectinga component of the DFT at the difference between twice the first inputfrequency and the second input frequency.
 26. A computer-readable mediumthat stores a computer program for analyzing distortion in a circuit,the computer program comprising instructions for: determining anoperating point for the circuit, the circuit having a linear offset anda nonlinear offset at the operating point; determining a first-ordersolution from the linear offset and an input having a first inputfrequency and a second input frequency; determining a harmonic componentof the nonlinear offset at a difference between the first inputfrequency and the second input frequency; and determining a second-ordersolution from the linear offset and the harmonic component of thenonlinear offset at the difference between the first input frequency andthe second input frequency, wherein the second-order solution providesan estimate for a second-order intermodulation product for the circuit.27. A computer-readable medium according to claim 26, wherein thecomputer program further includes instructions for: determining anestimate for a second-order intercept point for the circuit from thesecond-order solution and the first-order solution.
 28. Acomputer-readable medium according to claim 27, wherein determining theestimate for the second-order intercept point includes: calculating anintersection between a linear representation of the second-ordersolution and a linear representation of the first-order solution,wherein the linear representation of the second-order solution has aslope of about 2 dB/dB and the linear representation of the first-ordersolution has a slope of about 1 dB/dB.
 29. A computer-readable mediumaccording to claim 26, wherein the computer program further includesinstructions for calculating a DFT of the nonlinear offset, anddetermining the harmonic component of the nonlinear offset at thedifference between the first input frequency and the second inputfrequency includes selecting a component of the DFT at the differencebetween the first input frequency and the second input frequency,
 30. Acomputer-readable medium according to claim 26, wherein the computerprogram further includes instructions for: determining a harmoniccomponent of the nonlinear offset at twice the first input frequency;determining a second-order solution from the linear offset and theharmonic component of the nonlinear offset at twice the first inputfrequency; determining a harmonic component of the nonlinear offset at adifference between twice the first input frequency and the second inputfrequency; and determining a third-order solution from the linear offsetand the harmonic component of the nonlinear offset at the differencebetween twice the first input frequency and the second input frequency,wherein the third-order solution provides an estimate for a third-orderintermodulation product for the circuit.
 31. A computer-readable mediumaccording to claim 30, wherein the computer program further includesinstructions for: determining an estimate for a third-order interceptpoint for the circuit from the third-order solution and the and thefirst-order solution.
 32. A computer-readable medium according to claim31, wherein determining the estimate for the third-order intercept pointincludes: calculating an intersection between a linear representation ofthe third-order solution and a linear representation of the first-ordersolution, wherein the linear representation of the third-order solutionhas a slope of about 3 dB/dB and the linear representation of thefirst-order solution has a slope of about 1 dB/dB.
 33. Acomputer-readable medium according to claim 30, wherein the computerprogram further includes instructions for calculating a DFT of thenonlinear offset, determining the harmonic component of the nonlinearoffset at the difference between the first input frequency and thesecond input frequency includes selecting a component of the DFT at thedifference between the first input frequency and the second inputfrequency, determining the harmonic component of the nonlinear offset attwice the first input frequency includes selecting a component of theDFT at twice the first input frequency, and determining the harmoniccomponent of the nonlinear offset at the difference between twice thefirst input frequency and the second input frequency includes selectinga component of the DFT at the difference between twice the first inputfrequency and the second input frequency.